Note on the bias in the estimation of the serial correlation coefficient of AR(1) processes
Note on the bias in the estimation of the serial correlation coefficient of AR(1) processes
复制标题
注意 AR(1) 过程的序列相关系数估计中的偏差
DOI：
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发表时间：
2001
期刊：
影响因子：
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通讯作者：
M. Mudelsee
中科院分区：
文献类型：
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作者：
M. Mudelsee
本文导出了在自相关系数ρ的整个范围内实用的自相关估计量的均值和方差的近似公式。最小二乘估计量1
$$Sigma _{i = 1}^{n - 1} in _i in _{i = 1} /Sigma _{i = 1}^{n - 1} in _i^2 $$
对于均值已知的平稳AR（1）过程，研究了一种新的自回归模型。我们使用比率的二阶泰勒展开式，并使用算术-几何级数代替部分塞萨罗和。在平均数的情况下，我们推导出Marriott和Pope（1954）的公式，用（n - 1）-1代替（n）-1，并增加一个项<$（n - 1）-2。当ρ接近1时，这个新公式产生了预期的下降到零的负偏差。在方差Bartlett（1946）公式结果的情况下，用（n - 1）-1代替（n）-1。理论表达式与模拟实验相印证。比较表明，当ρ β> 0.88和n ≥ 20时，我们的平均值公式比白色（1961）的高阶近似更精确。原则上，所提出的方法可用于推导其他估计量和过程的近似公式。
AbstractWe derive approximating formulas for the mean and the variance of an autocorrelation estimator which are of practical use over the entire range of the autocorrelation coefficient ρ. The least-squares estimator 1
$$Sigma _{i = 1}^{n - 1} in _i in _{i = 1} /Sigma _{i = 1}^{n - 1} in _i^2 $$
is studied for a stationary AR(1) process with known mean. We use the second order Taylor expansion of a ratio, and employ the arithmetic—geometric series instead of replacing partial Cesaro sums. In case of the mean we derive Marriott and Pope’s (1954) formula, with (n - 1)-1 instead of (n)-1, and an additional term ∝ (n - 1)-2. This new formula produces the expected decline to zero negative bias as ρ approaches unity. In case of the variance Bartlett’s (1946) formula results, with (n — 1)-1 instead of (n)-1. The theoretical expressions are corroborated with a simulation experiment. A comparison shows that our formula for the mean is more accurate than the higher-order approximation of White (1961), for ¦ρ¦> 0.88 and n ≥ 20. In principal, the presented method can be used to derive approximating formulas for other estimators and processes.